Petersen and Tenner’s depth measures the distance of a permutation from its symmetric group’s identity and is known to be bound below by the average of two other measures of distance, length and reflection length. Permutations where this bound is an equality (i.e., permutations that are shallow) are known to not be characterizable by the containment and avoidance of patterns. However, using Hadjicostas and Monico’s recursive definition for shallowness, we show that the subset of shallow permutations which avoid 4231 can be characterized in this way. Such permutations are the ones which avoid 4231, 3412, 34521, 54123, 365214, 541632, 7652143, and 5476321.

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